NIMCET 2023 — Mathematics PYQ

Let A and B be two sets such that $A\cap X=B\cap X=\varnothing$ and $A\cup X=B\cup X$ for some set $X$.
To show: $A=B$

$A=A\cap (A\cup X)$

$=A\cap (A\cup X)(A\cup X=B\cup X)$

$=(A\cap B)\cup (A\cap X)$ (Distributive law)

$(A\cap B)\cup \varnothing ;(\because A\cap X=\varnothing )$

$=A\cap B\dots (i)$

Now, $B=B\cap (B\cup X)$

$=B\cap (A\cup X);(\because A\cup X=B\cup X)$

$=(B\cap A)\cup (B\cap X)$ (Distributive law)

$=(B\cap A)\cup \varnothing ;(\because B\cap X=\varnothing )$

$=B\cap A=A\cap B\dots (ii)$

Hence, from (i) and (ii), we get
$A=B$